The compact conformations of polymers are important because the native conformations of all bio-polymers with certain function are highly compact. The properties of mutil-contact bio-polymer chains were studied by Gaussian statistics of the random-flight chain. The theoretical expressions(were given, also), the calculations of probability distributions and correlation functions for different topologic cases were derived and made respectively. Comparison between single, double and triple contacts was also made. By means of setting the parameters, the results of the current calculations of the multiple contacts are just the same as those calculated by single, double or tripe contacts separately. It is a useful method to investigate native conformations of biopolymers. The probabilities of multi contacts and correlation functions between chains contacts were calculated for the Gaussian chains. Because the bond probability distributions are Gaussians distributions, the probability distributions of the separations of various points along the chains are always consecutive. All the contacts may break up into several groups, and each group consists of many contacts. Here we investigated the probability distribution from one group to three groups of contacts.